On the solution of a Riesz equilibrium problem and integral identities for special functions

The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. It seems that these identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler-Lagrange variational characterization, the Funk-Hecke formula, the Weyl regularity lemma, the maximum principle, and special properties of hypergeometric functions.Comment: Minor modifications. To appear in Journal of Mathematical Analysis and Applications (JMAA

A continuous minimax problem for calculating minimum norm polynomial interpolation points on the sphere

This paper considers the calculation of the minimum norm points for polynomial interpolation over the sphere S 2 ? R 3 . The norm of the interpolation operator ? n , considered as a map from C(S 2 ) to C(S 2 ), is given by ? ? n ? = max x ? S 2 ?B -1 b (x)? 1 , where the nonsingular matrix B and vector b are determined by the fundamental system of points x j ? S 2 , j = 1,?, d n . The problem is to choose the fundamental system to minimise ? ? n ?. Algorithms for solving this continuous minimax problem must be able to handle many local maxima close to the global maximum, and local maxima which lie close to each other along ridges. A first order dual algorithm is used to find a spherical parametrisation of a normalised fundamental system. The results suggest that for these points the growth in ? ? n ?, for n ? 30, is less than c 0 + c 1 n, where c 0 ? 1.8 and c 1 ? 0.7

Removing the mask -- reconstructing a scalar field on the sphere from a masked field

The paper analyses a spectral approach to reconstructing %the image of a scalar field on the sphere, given only information about a masked version of the field together with precise information about the (smooth) mask. The theory is developed for a general mask, and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree ($\ell \le 100$). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise

Extremal systems of points and numerical integration on the sphere

This paper considers extremal systems of points on the unit sphere S r ⊆ R r+1 , related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d n = dim P n points, where P n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S 2 of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments

Effect of Mesh Phasing on the Transmission Efficiency and Dynamic Performance of Wheel Hub Planetary Gear Sets

Transmission efficiency and refinement of planetary wheel hub gearing system are key design attributes for heavy and off-highway vehicles. Reduction of power loss, directly leading to the development of new generation ECO-axles requires analysis of gear contacting conditions for lubricated conjunctions to determine frictional performance. This is also affected by gear dynamics, which is a prerequisite for assessment of noise, vibration and harshness performance. Therefore, a combined tribo-dynamic analysis is essential. There is a dearth of such holistic analysis, particularly for the case of wheel hub planetary systems. The paper presents such an analysis, which has not hitherto been reported in literature. The inexorable interplay of transmission efficiency and noise, vibration and harshness refinement is demonstrated. The key attributes of noise, vibration and harshness refinement and transmission efficiency can pose contrary requirements and near-optimal conditions can be highlighted by mesh phasing of gearing contacts, thus alleviating the need for more complex gear teeth modifications entailing prohibitive manufacturing costs

Современные взгляды на ультразвуковую диагностику внутреннего эндометриоза

Представлен обзор данных о применении различных методик в диагностике эндометриоза. Показаны преимущества и высокая информативность трансвагинальной эхографии. Описаны эхографические критерии различных стадий эндометриоза. Отмечена связь эндометриоза с разными патологиями щитовидной и молочной желез.This paper reviews the data on the use of different techniques of endometriosis diagnosis. The advantages and high informativity of transvaginal ultrasonography are demonstrated. Ultrasonography criteria of different stages of endometriosis are described. Association of endometriosis and thyroid and breast pathology is demonstrated

Status of Muon Collider Research and Development and Future Plans

The status of the research on muon colliders is discussed and plans are outlined for future theoretical and experimental studies. Besides continued work on the parameters of a 3-4 and 0.5 TeV center-of-mass (CoM) energy collider, many studies are now concentrating on a machine near 0.1 TeV (CoM) that could be a factory for the s-channel production of Higgs particles. We discuss the research on the various components in such muon colliders, starting from the proton accelerator needed to generate pions from a heavy-Z target and proceeding through the phase rotation and decay ($\pi \to \mu \nu_{\mu}$) channel, muon cooling, acceleration, storage in a collider ring and the collider detector. We also present theoretical and experimental R & D plans for the next several years that should lead to a better understanding of the design and feasibility issues for all of the components. This report is an update of the progress on the R & D since the Feasibility Study of Muon Colliders presented at the Snowmass'96 Workshop [R. B. Palmer, A. Sessler and A. Tollestrup, Proceedings of the 1996 DPF/DPB Summer Study on High-Energy Physics (Stanford Linear Accelerator Center, Menlo Park, CA, 1997)].Comment: 95 pages, 75 figures. Submitted to Physical Review Special Topics, Accelerators and Beam